If you (like me) have trouble remembering “the trick” for proving Taylor’s Theorem (with the Lagrange form of the remainder), have a look at Timothy Gowers’s blog post on this, where you will find what is, perhaps, a more natural proof than the usual one.

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You can find a nice proof of Taylor’s Thm here: http://www.math.csusb.edu/faculty/pmclough/SPTT.pdf

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